Complete solution of parametrized Thue equations
Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
We consider the Lebesgue-Ramanujan-Nagell type equation , where and are unknown integers with . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all -integral points on a class of elliptic curves.
Comptage exact de discriminants d'extensions abéliennes
Le but de cet article est d’expliquer comment calculer exactement le nombre de classes d’isomorphismes d’extensions abéliennes de en degré inférieur ou égal à et de discriminant majoré par une borne donnée. On parvient par exemple à calculer le nombre de corps cubiques cycliques de discriminant inférieur ou égal à .
Computation of 2-groups of positive classes of exceptional number fields
We present an algorithm for computing the 2-group of the positive divisor classes in case the number field has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel in .
Computation of -adic heights and log convergence.
Computation of relative class numbers of imaginary abelian number fields.
Computation of the fundamental units and the regulator of a cyclic cubic function field.
Computation of the Selmer groups of certain parametrized elliptic curves
Computations of Galois representations associated to modular forms of level one
We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...
Computations with Witt vectors of length
In this paper we describe how to perform computations with Witt vectors of length in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the -invariant of the canonical lifting as a function on the -invariant of the ordinary elliptic curve in characteristic .
Computer-aided serendipity
Computer-free evaluation of an infinite double sum via Euler sums.
Computing all elements of given index in sextic fields with a cubic subfield
Computing all monogeneous mixed dihedral quartic extensions of a quadratic field
Let be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields with mixed signature having power integral bases and containing as a subfield. We also determine all generators of power integral bases in . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for
Computing fundamental domains for Fuchsian groups
We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group with cofinite area. As a consequence, we compute the invariants of , including an explicit finite presentation for .
Computing Igusa's local zeta functions of univariate polynomials, and linear feedback shift registers.
Computing integral points on Mordell's elliptic curves.
Computing Iwasawa modules of real quadratic number fields
Computing periods of cusp forms and modular elliptic curves.
Computing the modular degree of an elliptic curve.